Proposition 1 (Natural (\(T^f_N\) ) Inclusion Functions)  [1]}
Consider \({\color {black} \mathbb {I}}\mathcal {Z} \triangleq [\underline{z},\overline{z}] \in \mathbb {IR}^{n_z}\)  and \(f\triangleq [f_1,\dots , f_{n_x}]^\top :{\color {black} \mathcal {Z} \subset \mathbb {R}^{n_z}} \rightarrow \mathbb {R}^{n_x}\) , where each \(f_j, \ j\in {\color {black} \mathbb {N}_{n_x}}\) ,  is expressed as a finite composition of the operators \(+,-,\times ,/\)  and elementary functions (sine, cosine, exponential, square root,  etc.). A natural inclusion function \(T^f_N:\mathbb {IR}^{n_z} \rightarrow {\color {black} \mathbb {I}}\mathbb {R}^{n_x}\)  for \(f(\cdot )\)  is obtained by replacing each real variable \(z_i,i\in {\color {black} \mathbb {N}_{n_z}}\) , by its corresponding interval variable \([z_i] \triangleq {\color {black} \mathbb {I}}\mathcal {Z}_i=[\underline{z}_i,\overline{z}_i]\) , and each operator or function by its interval counterpart by applying interval arithmetic (cf. [1]}  for details).

